Lesson 17

Two Related Quantities, Part 1

Let’s use equations and graphs to describe relationships with ratios.

Which one would you choose? Be prepared to explain your reasoning.

A 5-pound jug of honey for $15.35
Three 1.5-pound jars of honey for $13.05

A 5-pound jug of honey labeled $15.35. Three 1.5-pound jars of honey.

Lin needs to mix a specific shade of orange paint for the set of the school play. The color uses 3 parts yellow for every 2 parts red.

Complete the table to show different combinations of red and yellow paint that will make the shade of orange Lin needs.

cups of red paint $(r)$	cups of yellow paint $(y)$	total cups of paint $(t)$
2	3
6
		20
	18
14
16
		50
	42

Lin notices that the number of cups of red paint is always $\frac25$ of the total number of cups. She writes the equation $r=\frac25 t$ to describe the relationship. Which is the independent variable? Which is the dependent variable? Explain how you know.
Write an equation that describes the relationship between $r$ and $y$ where $y$ is the independent variable.
Write an equation that describes the relationship between $y$ and $r$ where $r$ is the independent variable.
Use the points in the table to create two graphs that show the relationship between $r$ and $y$ . Match each relationship to one of the equations you wrote.

Are you ready for more?

A fruit stand sells apples, peaches, and tomatoes. Today, they sold 4 apples for every 5 peaches. They sold 2 peaches for every 3 tomatoes. They sold 132 pieces of fruit in total. How many of each fruit did they sell?

Equations are very useful for describing sets of equivalent ratios. Here is an example.

A pie recipe calls for 3 green apples for every 5 red apples. We can create a table to show some equivalent ratios.

We can see from the table that $r$ is always $\frac53$ as large as $g$ and that $g$ is always $\frac35$ as large as $r$ .

green apples ( $g$ )	red apples ( $r$ )
3	5
6	10
9	15
12	20

We can write equations to describe the relationship between $g$ and $r$ .

When we know the number of green apples and want to find the number of red apples, we can write:

$\displaystyle r=\frac53g$

In this equation, if $g$ changes, $r$ is affected by the change, so we refer to $g$ as the independent variable and $r$ as the dependent variable.

We can use this equation with any value of $g$ to find $r$ . If 270 green apples are used, then $\frac53 \boldcdot (270)$ or 450 red apples are used.

When we know the number of red apples and want to find the number of green apples, we can write:

$\displaystyle g=\frac35r$

In this equation, if $r$ changes, $g$ is affected by the change, so we refer to $r$ as the independent variable and $g$ as the dependent variable.

We can use this equation with any value of $r$ to find $g$ . If 275 red apples are used, then $\frac35 \boldcdot (275)$ or 165 green apples are used.

We can also graph the two equations we wrote to get a visual picture of the relationship between the two quantities.

Two graphs that represent a ratio of two quantities.

dependent variable

The dependent variable is the result of a calculation.

For example, a boat travels at a constant speed of 25 miles per hour. The equation $d=25t$ describes the relationship between the boat's distance and time. The dependent variable is the distance traveled, because $d$ is the result of multiplying 25 by $t$ .
independent variable

The independent variable is used to calculate the value of another variable.

For example, a boat travels at a constant speed of 25 miles per hour. The equation $d=25t$ describes the relationship between the boat's distance and time. The independent variable is time, because $t$ is multiplied by 25 to get $d$ .

Lesson 17

17.1: Which One Would You Choose?

17.2: Painting the Set

Summary

Glossary Entries